Prove that if the sum of a series converges absolutely...?

Question

then the absolute value of the sum of the series is < or = the sum of the absolute value of the series.

Answer 1

DC almost has it - just have to change a couple lines:

u_i ≤ |u_i|

becomes

-|u_i| ≤ u_i ≤ |u_i|

The rest changes similarly until:

Σ u_i ≤ Σ |u_i|

which becomes

-Σ |u_i| ≤ Σ u_i ≤ Σ |u_i|

which gives the statement |Σ u_i | ≤ Σ |u_i|.

DC did most of the work here, so I&#039;d give him &quot;best&quot; still, though.

Answer 2

it is by comparison test

u_i ≤ |u_i|

thus

Σ u_i ≤ Σ |u_i|

now, if the series converges absolutely that means .. . .

Σ |u_i| = M , for some real valued M

thus

Σ u_i ≤ Σ |u_i| = real M